A general topology-based framework for adaptive insertion of cohesive elements in finite element meshes

Large-scale simulation of separation phenomena in solids such as fracture, branching, and fragmentation requires a scalable data structure representation of the evolving model. Modeling of such phenomena can be successfully accomplished by means of cohesive models of fracture, which are versatile and effective tools for computational analysis. A common approach to insert cohesive elements in finite element meshes consists of adding discrete special interfaces (cohesive elements) between bulk elements. The insertion of cohesive elements along bulk element interfaces for fragmentation simulation imposes changes in the topology of the mesh. This paper presents a unified topology-based framework for supporting adaptive fragmentation simulations, being able to handle two- and three-dimensional models, with finite elements of any order. We represent the finite element model using a compact and “complete” topological data structure, which is capable of retrieving all adjacency relationships needed for the simulation. Moreover, we introduce a new topology-based algorithm that systematically classifies fractured facets (i.e., facets along which fracture has occurred). The algorithm follows a set of procedures that consistently perform all the topological changes needed to update the model. The proposed topology-based framework is general and ensures that the model representation remains always valid during fragmentation, even when very complex crack patterns are involved. The framework correctness and efficiency are illustrated by arbitrary insertion of cohesive elements in various finite element meshes of self-similar geometries, including both two- and three-dimensional models. These computational tests clearly show linear scaling in time, which is a key feature of the present data-structure representation. The effectiveness of the proposed approach is also demonstrated by dynamic fracture analysis through finite element simulations of actual engineering problems.

ParTopS: compact topological framework for parallel fragmentation simulations

Cohesive models are used for simulation of fracture, branching and fragmentation phenomena at various scales. Those models require high levels of mesh refinement at the crack tip region so that nonlinear behavior can be captured and physical results obtained. This imposes the use of large meshes that usually result in computational and memory costs prohibitively expensive for a single traditional workstation. If an extrinsic cohesive model is to be used, support for dynamic insertion of cohesive elements is also required. This paper proposes a topological framework for supporting parallel adaptive fragmentation simulations that provides operations for dynamic insertion of cohesive elements, in a uniform way, for both two- and three-dimensional unstructured meshes. Cohesive elements are truly represented and are treated like any other regular element. The framework is built as an extension of a compact adjacency-based serial topological data structure, which can natively handle the representation of cohesive elements. Symmetrical modifications of duplicated entities are used to reduce the communication of topological changes among mesh partitions and also to avoid the use of locks. The correctness and efficiency of the proposed framework are demonstrated by a series of arbitrary insertions of cohesive elements into some sample meshes.     

An open source program to generate zero-thickness cohesive interface elements

An open source program to generate zero-thickness cohesive interface elements in existing finite element discretizations is presented. This contribution fills the gap in the literature that, to the best of the author’s knowledge, there is no such program exists. The program is useful in numerical modeling of material/structure failure using cohesive interface elements. The program is able to generate one/two dimensional, linear/quadratic cohesive elements (i) at all inter-element boundaries, (ii) at material interfaces and (iii) at grain boundaries in polycrystalline materials. Algorithms and utilization of the program is discussed. Several two dimensional and three dimensional fracture mechanics problems are given including debonding process of material interfaces, multiple delamination of composite structures, crack propagation in polycrystalline structures.     

Parallel Simulations of Dynamic Fracture Using Extrinsic Cohesive Elements

In this paper, we present a novel parallel implementation of extrinsic initially rigid cohesive elements in an explicit finite element solver designed for the simulation of dynamic fracture events. The implementation is based on activating instead of inserting the cohesive elements and uses ParFUM, a parallel framework specifically developed for simulations involving unstructured meshes. Aspects of the parallel implementation are described, along with an analysis of its performance on 1 to 512 processors. Important cache effects and communication costs are included in this analysis. The implementation is validated by simulating the trapping of a crack along an inclined material interface.     

A cohesive approach to thin-shell fracture and fragmentation

We develop a finite-element method for the simulation of dynamic fracture and fragmentation of thin-shells. The shell is spatially discretized with subdivision shell elements and the fracture along the element edges is modeled with a cohesive law. In order to follow the propagation and branching of cracks, subdivision shell elements are pre-fractured ab initio and the crack opening is constrained prior to crack nucleation. This approach allows for shell fracture in an in-plane tearing mode, a shearing mode, or a bending of hinge mode. The good performance of the method is demonstrated through the simulation of petalling failure experiments in aluminum plates.     

A one field full discontinuous Galerkin method for Kirchhoff–Love shells applied to fracture mechanics

In order to model fracture, the cohesive zone method can be coupled in a very efficient way with the finite element method. Nevertheless, there are some drawbacks with the classical insertion of cohesive elements. It is well known that, on one the hand, if these elements are present before fracture there is a modification of the structure stiffness, and that, on the other hand, their insertion during the simulation requires very complex implementation, especially with parallel codes. These drawbacks can be avoided by combining the cohesive method with the use of a discontinuous Galerkin formulation. In such a formulation, all the elements are discontinuous and the continuity is weakly ensured in a stable and consistent way by inserting extra terms on the boundary of elements. The recourse to interface elements allows to substitute them by cohesive elements at the onset of fracture.The purpose of this paper is to develop this formulation for Kirchhoff–Love plates and shells. It is achieved by the establishment of a full DG formulation of shell combined with a cohesive model, which is adapted to the special thickness discretization of the shell formulation. In fact, this cohesive model is applied on resulting reduced stresses which are the basis of thin structures formulations. Finally, numerical examples demonstrate the efficiency of the method.     

Simulation of delamination in stringer stiffened fiber-reinforced composite shells

Fiber-reinforced composites are often used for high performance lightweight structures. For an enhanced exploitation of material reserves, fracture mechanisms should be taken into consideration. In this work, delamination and skin-stringer separation are examined in the framework of the finite element method. A cohesive interface element is used which is written in stress-strain relationships. The cohesive law rests upon a Smith-Ferrante type free energy function. It is edited so that only tensile normal or shear stresses provoke damage and contact is accounted for by an additional penalty term. Some numerical examples show the applicability of the proposed model.     

Breaking processes in three-dimensional bonded granular materials with general shapes

The paper presents an extension to the spheropolyhedra method for the simulation of granular materials comprising particles of general shapes with bonding. A bonding, cement, or cohesion model for particles sharing common faces is introduced. The bonding force is elastic and has a strain-based breaking threshold for modelling fracture. An initial study is conducted based on the Brazilian tensile test to check how the parameters of the proposed model affect the principal variables measured in this test. Afterwards, solid cubic blocks are then subjected to a triaxial test to explore the mathematical macroscopic failure model. It is found that the peak strength envelope is the product of the superposition of frictional and fracture failure mechanisms. The fracture failure is mainly produced by an avalanche of broken cohesive bonds. The intensity of the avalanche exhibits a power law distribution, as reported in previous studies. The method allows for random divisions of solid bodies without any pre-existing internal voids. It offers a natural, effective tool to model, simulate and study fragmentation processes in 3D.     

Solid modeling aspects of three-dimensional fragmentation

The ways in which the topology and geometry of a three-dimensional finite-element model may evolve as a consequence of fracture and fragmentation are enumerated, and the actions which may be taken in order to update the boundary representation of the solid so as to faithfully reflect that evolution are described. Arbitrary topological and geometrical evolution of a three-dimensional solid, not necessarily restricted to an evolution of its surface, are addressed. Solids are described by their boundary representation (BRep) and a surface and volume triangulation. Fracture processes are modeled by the introduction of cohesive elements at element interfaces. Simple rules are shown to enable the simulation of strikingly complex crack patterns. The scope and versatility of the approach is illustrated with the aid of selected examples of application.     

OBSIFRAC: database-supported software for 3D modeling of rock mass fragmentation

Under stress, fractures in rock masses tend to form fully connected networks. The mass can thus be thought of as a 3D series of blocks produced by fragmentation processes. A numerical model has been developed that uses a relational database to describe such a mass. The model, which assumes the fractures to be plane, allows data from natural networks to test theories concerning fragmentation processes. In the model, blocks are bordered by faces that are composed of edges and vertices. A fracture can originate from a seed point, its orientation being controlled by the stress field specified by an orientation matrix. Alternatively, it can be generated from a discrete set of given orientations and positions. Both kinds of fracture can occur together in a model. From an original simple block, a given fracture produces two simple polyhedral blocks, and the original block becomes compound. Compound and simple blocks created throughout fragmentation are stored in the database. Several fragmentation processes have been studied. In one scenario, a constant proportion of blocks is fragmented at each step of the process. The resulting distribution appears to be fractal, although seed points are random in each fragmented block. In a second scenario, division affects only one random block at each stage of the process, and gives a Weibull volume distribution law. This software can be used for a large number of other applications.     

A Procedure for Tetrahedral Boundary Layer Mesh Generation

. We develop a methodology for introducing regions of high anisotropy in existing isotropic unstructured grids in complex, curved, three-dimensional domains. The new procedures are here applied to the capturing of solution features in the proximity of model boundaries (e.g. boundary layers). Suitable voids are created in an existing grid in the regions of localization using a mesh motion algorithm that solves a fictitious elasticity problem. The voids are then filled with stacks of prisms that are subsequently tetrahedronized to yield a simplicial mesh. The mesh motion algorithm allows us to deal in a simple and effective manner with the problem of self-intersection of elements in concave regions of the model boundaries, and in the case of closely spaced model faces, avoiding the need for cross-over checks and complex grid correction procedures. The capabilities and performance of the proposed methodology are illustrated with the help of practical examples.     

Functions Defining Arbitrary Meshes – A Flexible Interface between Numerical Data and Visualization Routines

Most of the rendering tools in scientific visualization are restricted to special data structures which differ substantially from the data formats used in numerical applications. Trying to close this gap, we present an interface between data from numerical methods on general types of grids - like cuboidal, prismatic, simplicial, parametric, mixed, or hierarchical meshes — and general visualization routines. It is based on a procedural approach managing a collection of arbitrary elements and a set of functions describing each element type. No mapping of (an in general enormous amount of) numerical data onto new data structures is necessary; a user may use his own data structures and only has to provide this small set of procedures and functions. The visualization tools will then use these routines to access (temporarily and locally) data of interest, like information about a single element. Compared with display routines on a specialized data structure, this general interface does not produce much cpu overhead.     

Iterationsverfahren bei der konformen Abbildung

Zusammenfassung Bei der konformen Abbildung des Inneren des Einheitskreises auf ein einfach zusammenh?ngendes Gebiet nach der Methode vonTheodorsen tritt eine singul?re nichtlineare Integralgleichung auf. Durch Diskretisation entsteht daraus ein nichtlineares Gleichungssystem, mit dessen iterativer L?sung wir uns besch?ftigen. Es wird ein Satz bewiesen, der besagt, da? das Einzelschrittverfahren doppelt so schnell konvergiert wie das Gesamtschrittverfahren (für das ein Konvergenzsatz vonOpitz, Ostrowski undSaltzer bewiesen wurde). Im n?chsten Abschnitt wenden wir das SOR-Verfahren (Successive Over-Relaxation) auf die beimNewton-Verfahren in jedem Schritt entstehenden linearen Gleichungssysteme an; der Rechenaufwand kann, verglichen mit direkten Verfahren, besonders bei gro?en Systemen stark reduziert werden. Schlie?lich wird vorgeschlagen, das SOR-Verfahren direkt beim nichtlinearen System anzuwenden. Vermutungen über die “Konvergenzgrenze” und den “optimalen Relaxationsfaktor” werden durch ein numerisches Beispiel erh?rtet.

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Summary We consider the conformal mapping of the interior of the unit circle onto the interior of a general closed curve by the method ofTheodorsen. There is to solve a singular nonlinear integral equation. By discretisation one gets a system of nonlinear equations; we examine different iterative methods of solving this system. A theorem is proved which says that the single step iteration converges twice as fast as the total step iteration (on the latter a convergence theorem ofOpitz, Ostrowski andSaltzer is known). In the next section we apply the SOR-method (Successive Over-Relaxation) to the systems of linear equations which appear at every step of theNewton-method. Compared with direct methods the computational work is reduced, especially on great systems. Finally we propose to apply the SOR-method directly to the nonlinear system. Conjetures on the “limit of convergence” and the “optimal relaxation factor” are confirmed by an example.

     

A hierarchical spatial data structure for global geographic information systems

Hierarchical spatial data structures offer the distinct advantages of data compression and fast access, but are difficult to adapt to the globe. Following Dutton, we propose projecting the globe onto an octahedron and then recursively subdividing each of its eight triangular faces into four triangles. We provide procedures for addressing the hierarchy and for computing addresses in the hierarchical structure from latitude and longitude and vice versa. At any level in the hierarchy the finite elements are all triangles, but are only approximately equal in area and shape; we provide methods for computing area and for finding the addresses of neighboring triangles.     

基于内聚力模型的界面破坏分析

界面破坏是材料与结构失效的常见形式,准确分析模拟界面损伤演化和最终破坏对评估材料乃至结构性能至关重要.在简要介绍内聚力模型的基础上,用Abaqus中的内聚力单元分别对均质材料和非均质材料界面破坏过程进行模拟,数值结果与理论结果吻合良好,表明内聚力单元适用于材料界面破坏分析.     

Combining predicate and numeric abstraction for software model checking

Predicate (PA) and numeric (NA) abstractions are the two principal techniques for software analysis. In this paper, we develop an approach to couple the two techniques tightly into a unified framework via a single abstract domain called NumPredDom. In particular, we develop and evaluate four data structures that implement NumPredDom but differ in their expressivity and internal representation and algorithms. All our data structures combine BDDs (for efficient propositional reasoning) with data structures for representing numerical constraints. Our technique is distinguished by its support for complex transfer functions that allow two-way interaction between predicate and numeric information during state transformation. We have implemented a general framework for reachability analysis of C programs on top of our four data structures. Our experiments on non-trivial examples show that our proposed combination of PA and NA is more powerful and more efficient than either technique alone.     

Mixed Dimensional Coupling in Finite Element Stress Analysis

Many analysis models utilize finite elements of reduced dimension. However, to capture stress concentrations at local details, it would be desirable to combine the reduced dimensional element types with higher dimensional elements in a single finite element model. It is therefore important in such cases to integrate into the analyses some scheme for coupling the element types that conforms to the governing equations of the problem. In this paper, a novel method that can correctly couple beams to solids, beams to shells and shells to solids for elastic problems is presented. The approach adopted is to equate the work done on either side of the interface between dimensions, and this leads to multi-point constraint equations, thus providing a relationship among nodal degrees of freedom between the differing element types. Example results show that the proposed technique does not introduce any spurious stresses at the dimensional interfaces. ID="A1" Correspondence and offprint requests to: C. G. Armstrong, School of Mechanical and Manufacturing Engineering, The Queen's University of Belfast, Ashby Building, Stranmillis Road, Belfast BT9 5AH, Northern Ireland. E-mail: c.armstrong@qub.ac.uk     

Bi-Objective Optimization Method for Horizontal Fragmentation Problem in Relational Data Warehouses as a Linear Programming Problem

ABSTRACT

In this work, we relied on a particular exact method to solve NP-Hard problem of determining a horizontal fragmentation scheme in relational data warehouses. The method used is that of linear programming which is distinguished by other methods by the existence of practical methods that facilitate the resolution of problems that may be described in linear form. We quote the Simplex method and the interior points. To meet the linearity of the objective function and constraints, we used initially “De Morgan” theorem, which is based on properties of sets to transform and optimize decision queries, from any form to a linear one.

In addition to designing and solving the selection problem of horizontal fragmentation technique, we considered the problem in two simultaneous objectives, namely: the number of Inputs/Outputs needed to run the global workload, and number of fragments generated to identify the best solutions compared to the concept of Pareto dominance.

In addition, to carry out our work, we used the Benchmark APB1 invoked by a workload, to achieve satisfactory results.